handout6 - Handout 6 Electrons in Periodic Potentials In...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 6 Electrons in Periodic Potentials In this lecture you will learn: • Bloch’s theorem and Bloch functions • Electron Bragg scattering and opening of bandgaps • Free electron bands and zone folding • Energy bands in 1D, 2D, and 3D lattices ECE 407 – Spring 2009 – Farhan Rana – Cornell University Atomic Potentials in Crystals The potential energy of an electron due to a single isolated atom looks like: 0 x ( ) r V r In a crystal, the potential energy due to all the atoms in the lattice looks like: 0 x ( ) r V r Energy levels The lowest energy levels and wavefunctions of electrons remain unchanged when going from an isolated atom to a crystal The higher energy levels (usually corresponding to the outermost atomic shell) get modified, and the corresponding wavefunctions are no longer localized at individual atoms but become spread over the entire crystal Potential of an isolated atom Potential in a crystal 0 0 Energy levels Vacuum level
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Properties of Atomic Potentials in Crystals • The atomic potential is lattice periodic (even for a lattice with a basis): ( ) ( ) r V R r V r r r = + where is any lattice vector R r • Because the atomic potential is lattice periodic, it can be written as a convolution (assuming a lattice in “ d ” dimensions) and expanded in a Fourier series of the type: ( ) ( ) r G i j d j j e G V r V r r r r . = ( ) ( ) ( ) = j j d R r r V r V r r r r δ where only the reciprocal lattice vectors appear in the exponential The Fourier components of the periodic potential contain only the reciprocal lattice vectors ( ) cell primitive one in potential = r V r ( ) ( ) r V R r V r r r = + Verify that: ECE 407 – Spring 2009 – Farhan Rana – Cornell University Properties of Electron Wavefunctions in Crystals Electrons in a crystal satisfy the Schrodinger equation: ( ) ( ) ( ) ( ) r E r r V r m r r r r h ψ ψ ψ = + 2 2 2 Where: ( ) ( ) r V R r V r r r = + Since the potential is periodic, and one lattice site is no different than any other lattice site, the solutions must satisfy: ( ) ( ) 2 2 r R r r r r ψ ψ = + This implies that the wavefunction at positions separated by a lattice vector can only differ by a phase factor: ( ) ( ) ( ) r e R r R i r r r r ψ ψ θ = + It follows that both the following relations must hold: ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) r e R R r r e R r e R R r R R i R R i R i r r r r r r r r r r r r r r r ψ ψ ψ ψ ψ θ θ θ θ ' ' ' ' ' + + = + + = + = + + Which implies: ( ) ( ) ( ) ' ' R R R R r r r r + = + θ θ θ
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Properties of Electron Wavefunctions in Crystals The simplest, and the only way, that the relation: can hold for all lattice vectors is if the phase is a linear scalar function of the vector : ( ) ( ) ( ) ' ' R R R R r r r r + = + θ θ θ R r ( ) R k R r r r . = θ where is some vector. It follows that our solutions must satisfy: k r ( ) ( ) r e R r R k i r r r r r ψ ψ .
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